Digital-to-analog conversion systems are subject to errors that result in a difference between the analog output signal output by the digital-to-analog conversion system and the analog signal represented by the digital signal input to the digital-to-analog conversion system. In some digital-to-analog conversion systems, the error to which the digital-to-analog conversion system is subject is a periodic error having a periodicity equal to that of an N-th sub-harmonic of the conversion frequency Fs of the digital-to-analog conversion system, i.e., the frequency at which the digital-to-analog conversion system converts successive values of the digital input signal to respective segments of the analog output signal. The error is periodic in the sense that the error varies from period-to-period of the conversion frequency in a repetitive pattern. The repetitive pattern has a periodicity equal to that of the above-mentioned sub-harmonic of the conversion frequency. In many cases, the digital-to-analog conversion system is subject to more than one periodic error. In this case, the description of periodic error herein applies to each periodic error to which the digital-to-analog conversion system is subject.
Some digital-to-analog conversion systems are composed of a digital-to-analog converter and one or more auxiliary circuits. The auxiliary circuits operate in response to clock signals at a sub-harmonic of the conversion frequency of the digital-to-analog converter. Leakage of signals from such auxiliary circuits into the analog signal path or clock signal path of the digital-to-analog converter subjects the digital-to-analog conversion system to a periodic error having a periodicity equal to that of a sub-harmonic of the conversion frequency of the digital-to-analog conversion system. Additionally or alternatively, leakage of signals from nearby circuits not part of the digital-to-analog conversion system but that operate in response to a clock signal at a sub-harmonic of the conversion frequency of the digital-to-analog conversion system can subject the digital-to-analog conversion system to a periodic error having a periodicity equal to that of the sub-harmonic of the conversion frequency.
A periodic leakage signal leaking into the digital-to-analog conversion system can appear directly in the analog output signal output by the digital-to-analog conversion system. In this case, the digital-to-analog conversion system can be regarded as being subject to a periodic offset error. A periodic offset error is an example of a signal-independent periodic error as it is independent of the analog signal represented by the digital input signal input to the digital-to-analog conversion system. The analog signal represented by the digital input signal will be referred to as a first analog signal.
Additionally or alternatively, a periodic leakage signal leaking into the digital-to-analog conversion system can modulate the gain of the digital-to-analog conversion system by modulating a bias signal or a supply voltage. In this case, the digital-to-analog conversion system can be regarded as being subject to a periodic gain error. Additionally or alternatively, such periodic leakage signal can modify the timing of transitions of the analog output signal generated by the digital-to-analog conversion system from one conversion period to the next. In this case, the digital-to-analog conversion system can be regarded as being subject to a periodic timing error. A periodic gain error and a periodic timing error are examples of signal-dependent periodic errors. A signal-dependent periodic error is a periodic error that depends on the first analog signal. A digital-to-analog conversion system may be subject to either or both a signal-independent periodic error and a signal-dependent error. Other types of periodic error are possible.
Other digital-to-analog conversion systems are composed of interleaved digital-to-analog converters. Each digital-to-analog converter operates at a conversion frequency of Fs/N, where N is the number of digital-to-analog converters constituting the digital-to-analog conversion system. Mismatches of timing and conversion gain, which may be frequency-dependent, among the digital-to-analog converters subject the digital-to-analog conversion system to a periodic error having a periodicity equal to that of a sub-harmonic of the conversion frequency of the digital-to-analog conversion system. The mismatches of timing and conversion gain can be caused by differences among the constituent digital-to-analog converters. Additionally or alternatively, such mismatches can be caused by leakage signals from nearby circuits not part of the digital-to-analog conversion system but that operate in response to a clock signal at a sub-harmonic of the conversion frequency of the digital-to-analog conversion system. Such mismatches of timing and conversion gain can be regarded as subjecting the digital-to-analog conversion system to periodic offset, gain and timing errors similar to those described above.
In a digital-to-analog conversion system subject to a signal-independent periodic error, such as a periodic offset error, the signal-independent error appears directly in the analog output signal output by the digital-to-analog conversion system. As a result, the analog output signal differs dynamically from the first analog signal by an error equal to the signal-independent periodic error. The error in the analog output signal has a periodicity equal to that of the sub-harmonic of the conversion frequency.
In a digital-to-analog conversion system subject to a signal-dependent periodic error, such as a periodic gain error or a periodic timing error, the spectrum of the signal-dependent periodic error is convolved with the spectrum of the first analog signal (gain error), or the time derivative of the spectrum of the first analog signal (timing error). As a result, the analog output signal differs dynamically from the first analog signal by an error composed of unwanted images of the first analog signal. For a first analog signal having a frequency of fin and a digital-to-analog conversion system subject to a periodic signal-dependent error having a frequency of Fs/N, the unwanted images of the first analog signal have various amplitudes and are at frequencies of fin±MFs/N, where M is an integer. Since the error in the analog output signal depends on the first analog signal, or its time derivative, any periodicity in the error in the analog output signal will depend on the periodicity of the first analog signal and the periodicity of the periodic error.
The digital-to-analog conversion system being subject to a periodic error having a periodicity equal to that of a sub-harmonic of the conversion frequency causes an error in the analog output signal output by the digital-to-analog conversion system, and therefore impairs the conversion accuracy of the digital-to-analog conversion system. Such impairment is undesirable. Accordingly, what is needed is a way to mitigate the periodic errors to which digital-to-analog conversion systems are subject.